Conductor of an abelian group

Abstract

Let E ( G ) = End ( G ) / N ( End ( G ) ) . Our goal in this paper is to study direct sum decompositions of certain reduced torsion-free finite rank ( rtffr) abelian groups by introducing an ideal τ of E ( G ) called a conductor of G. This ideal induces a natural ring decomposition E ( G ) = E ( G ) ( τ ) × E ( G ) τ and a natural direct sum decomposition G = G ( τ ) ⊕ G τ for an rtffr group G. Let { G 1 , … , G t } be a set of strongly indecomposable rtffr groups such that G i ≇ ˙ G j for each i ≠ j ∈ { 1 , … , t } , and such that E ( G i ) is a Dedekind domain for each i ∈ { 1 , … , t } . Let n 1 , … , n t > 0 be integers and let G ¯ = G 1 n 1 ⊕ ⋯ ⊕ G t n t . We say that G has semi-primary index in G ¯ if for each i = 1 , … , t there is a primary ideal P i ⊂ End ( G i ) such that P 1 G 1 n 1 ⊕ ⋯ ⊕ P t G t n t ⊂ G ⊂ G ¯ . The group G is balanced in G ¯ if G ⊂ G ¯ and if E ( G ) ⊂ E ( G ¯ ) . We say that G is a balanced semi-primary group if there is a balanced embedding G ⊂ G ¯ such that G has semi-primary index in G ¯ . Theorem If G is a balanced semi-primary rtffr group then G has a locally unique decomposition and the local refinement property.